Linear Transformations on Algebras of Matrices: The Invariance of the Elementary Symmetric Functions

1959 ◽  
Vol 11 ◽  
pp. 383-396 ◽  
Author(s):  
Marvin Marcus ◽  
Roger Purves
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Similarity Transformation ◽  
Elementary Symmetric Function ◽  
Linear Transformations ◽  
Elementary Symmetric Functions ◽  
Image Position

In this paper we examine the structure of certain linear transformations T on the algebra of w-square matrices Mn into itself. In particular if A ∈ Mn let Er(A) be the rth elementary symmetric function of the eigenvalues of A. Our main result states that if 4 ≤ r ≤ n — 1 and Er(T(A)) = Er(A) for A ∈ Mn then T is essentially (modulo taking the transpose and multiplying by a constant) a similarity transformation:No such result as this is true for r = 1,2 and we shall exhibit certain classes of counterexamples. These counterexamples fail to work for r = 3 and the structure of those T such that E3(T(A)) = E3(A) for all ∈ Mn is unknown to us.

An Inequality for Elementary Symmetric Functions

1972 ◽  
Vol 15 (1) ◽  
pp. 133-135 ◽  
Author(s):  
K. V. Menon
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Geometric Mean ◽  
Binomial Coefficient ◽  
Elementary Symmetric Function ◽  
Elementary Symmetric Functions ◽  
Image Position

Let Er denote the rth elementary symmetric function on α1 α2,…,αm which is defined by1E0 = 1 and Er=0(r>m).We define the rth symmetric mean by2where denote the binomial coefficient. If α1 α2,…,αm are positive reals thenwe have two well-known inequalities3and4In this paper we consider a generalization of these inequalities. The inequality (4) is known as Newton's inequality which contains the arithmetic and geometric mean inequality.

Linear Transformations on Matrices: the Invariance of the Third Elementary Symmetric Function

1970 ◽  
Vol 22 (4) ◽  
pp. 746-752 ◽  
Author(s):  
Leroy B. Beasley
Keyword(s):  
Linear Transformation ◽  
Symmetric Function ◽  
Similarity Transformation ◽  
Elementary Symmetric Function ◽  
Complex Numbers ◽  
Linear Transformations ◽  
The Third ◽  
Image Position

Let T be a linear transformation on Mn the set of all n × n matrices over the field of complex numbers, . Let A ∈ Mn have eigenvalues λ1, …, λn and let Er(A) denote the rth elementary symmetric function of the eigenvalues of A :Equivalently, Er(A) is the sum of all the principal r × r subdeterminants of A. T is said to preserve Er if Er[T(A)] = Er(A) for all A ∈ Mn. Marcus and Purves [3, Theorem 3.1] showed that for r ≧ 4, if T preserves Er then T is essentially a similarity transformation; that is, either T: A → UAV for all A ∈ Mn or T: A → UAtV for all A ∈ Mn, where UV = eiθIn, rθ ≡ 0 (mod 2π).

Inequalities for Generalized Symmetric Functions

1969 ◽  
Vol 12 (5) ◽  
pp. 615-623 ◽  
Author(s):  
K.V. Menon
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Symmetric Function ◽  
Generating Series ◽  
Image Position ◽  
Complete Symmetric Function

The generating series for the elementary symmetric function Er, the complete symmetric function Hr, are defined byrespectively.

scholarly journals The asymptotic behavior of elementary symmetric functions on a probability distribution

2001 ◽  
Vol 14 (3) ◽  
pp. 237-248 ◽  
Author(s):  
V. S. Kozyakin ◽  
A. V. Pokrovskii
Keyword(s):  
Asymptotic Behavior ◽  
Probability Distribution ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Symmetric Function ◽  
Elementary Symmetric Functions ◽  
Random Mappings

The problem on asymptotic of the value π(m,n)=m!σm(p(1,n),p(2,n),…,p(n,n)) is considered, where σm(x1,x2,…,xn) is the mth elementary symmetric function of n variables. The result is interpreted in the context of nonequiprobable random mappings theory.

Linear Transformations on Matrices: The Invariance of a Class of General Matrix Functions

1967 ◽  
Vol 19 ◽  
pp. 281-290 ◽  
Author(s):  
E. P. Botta
Keyword(s):  
Vector Space ◽  
Symmetric Function ◽  
Elementary Symmetric Function ◽  
Matrix Functions ◽  
Linear Transformations ◽  
General Matrix ◽  
Linear Maps ◽  
Generalized Matrix ◽  
Image Position

Let Mm(F) be the vector space of m-square matriceswhere F is a field; let f be a function on Mm(F) to some set R. It is of interest to determine the linear maps T: Mm(F) → Mm(F) which preserve the values of the function ƒ; i.e., ƒ(T(X)) = ƒ(X) for all X. For example, if we take ƒ(X) to be the rank of X, we are asking for a determination of the types of linear operations on matrices that preserve rank. Other classical invariants that may be taken for f are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues. Dieudonné (1), Hua (2), Jacobs (3), Marcus (4, 6, 8), Mori ta (9), and Moyls (6) have conducted extensive research in this area. A class of matrix functions that have recently aroused considerable interest (4; 7) is the generalized matrix functions in the sense of I. Schur (10).

scholarly journals On Determinants of Symmetric Functions

1927 ◽  
Vol 1 (1) ◽  
pp. 55-61 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
The Other ◽  
Elementary Symmetric Functions ◽  
The Difference ◽  
Image Position

The result of dividing the alternant |aαbβcγ…| by the simplest alternant |a0b1c2…| (the difference-product of a, b, c, …) is known to be a symmetric function expressible in two distinct ways, (1) as a determinant having for elements the elementary symmetric functions C, of a, b, c, …, (2) as a determinant having for elements the complete homogeneous symmetric functions Hr. For exampleThe formation of the (historically earlier) H-determinant is evident. The suffixes in the first row are the indices of the alternant; those of the other rows decrease by unit steps. This result is due to Jacobi.

scholarly journals Lyndon Words and Transition Matrices between Elementary, Homogeneous and Monomial Symmetric Functions

10.37236/1044 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Andrius Kulikauskas ◽  
Jeffrey Remmel
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Symmetric Function ◽  
Transition Matrices ◽  
Elementary Symmetric Functions ◽  
Combinatorial Interpretations ◽  
Lyndon Words

Let $h_\lambda$, $e_\lambda$, and $m_\lambda$ denote the homogeneous symmetric function, the elementary symmetric function and the monomial symmetric function associated with the partition $\lambda$ respectively. We give combinatorial interpretations for the coefficients that arise in expanding $m_\lambda$ in terms of homogeneous symmetric functions and the elementary symmetric functions. Such coefficients are interpreted in terms of certain classes of bi-brick permutations. The theory of Lyndon words is shown to play an important role in our interpretations.

Probability and the elementary symmetric functions

1973 ◽  
Vol 74 (1) ◽  
pp. 133-139 ◽  
Author(s):  
J. Denmead Smith
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Random Variables ◽  
Elementary Symmetric Function ◽  
Independent Random Variables ◽  
Elementary Symmetric Functions ◽  
Congruent Modulo

Let p be a prime, and suppose that x1,…,xN are independent random variables which take the values 0, 1,…,p − 1 with probabilities s0, sl…,sp−1 where s0+…+sp−1 = 1 and 0 < sk < 1 for each k. PN(n) denotes the probability that the elementary symmetric function σr(x1,…,xN) = ∑x1…,xr of the rth degree in the variables x1,…,xN is congruent, modulo p, to a prescribed integer n.

Two inequalities for the complete symmetric functions

1978 ◽  
Vol 84 (1) ◽  
pp. 1-3 ◽  
Author(s):  
V. J. Baston
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Positive Definite ◽  
Even Order ◽  
Image Position ◽  
Complete Symmetric Function ◽  
Special Case

In (l) Hunter proved that the complete symmetric functions of even order are positive definite by obtaining the inequalitywhere ht denotes the complete symmetric function of order t. In this note we show that the inequality can be strengthened, which, in turn, enables theorem 2 of (l) to be sharpened. We also obtain a special case of an inequality conjectured by McLeod(2).

Equally Inclined Spheres

1962 ◽  
Vol 58 (2) ◽  
pp. 420-421 ◽  
Author(s):  
J. G. Mauldon
Keyword(s):  
Symmetric Functions ◽  
Linear Equations ◽  
Quadratic Equation ◽  
Cartesian Coordinates ◽  
Elementary Symmetric Functions ◽  
Equation Solving ◽  
Image Position

If five spheres in 3-space are such that each pair is inclined at the same non-zero angle θ, then where b1, …, b5 (the ‘bends’ (1) of the spheres) are the reciprocals of their radii. To prove this result, establish a system of rectangular cartesian coordinates (x, y, z) and let the spheres have centres (xi, yi, zi) and radii , where i = 1,…, 5. Then for x5, y5, z5, r5 we have the equations which, on subtraction, yield three linear equations and one quadratic equation. Solving the three linear equations for x5, y5, z5 and substituting, we see that the required relation is algebraic (indeed quadratic) in r5 and hence in b5. Since it is also symmetric in b1,…, b5, it follows that it can be expressed as a polynomial relation in the elementary symmetric functions p1, …, p5 in b5, …, b5.

Sign in / Sign up

Export Citation Format

Share Document